Axial Nucleon and Nucleon to Delta form fractors and the Goldberger-Treiman Relations from Lattice QCD

TL;DR

This study uses lattice QCD to compute nucleon and nucleon-to-Delta form factors, testing the Goldberger-Treiman relations and confirming their validity across different quark masses and computational methods.

Contribution

It provides the first comprehensive lattice QCD evaluation of both nucleon and N to Delta transition form factors, validating the Goldberger-Treiman relations.

Findings

Ratios $G_{ ext{pi}N ext{Delta}}(q^2)/G_{ ext{pi}NN}(q^2)$ and $2C_5^A(q^2)/G_A(q^2)$ are constant with respect to $q^2$.

The ratios are approximately 1.61, confirming the Goldberger-Treiman relations.

Form factors are consistent across quenched and dynamical fermion calculations.

Abstract

We evaluate the nucleon axial form factor, $G_A(q^2)$, and induced pseudoscalar form factor, $G_p(q^2)$, as well as the pion-nucleon form factor, $G_{\pi N N}(q^2)$, in lattice QCD. We also evaluate the corresponding nucleon to $\Delta$ transition form factors, $C_5^A(q^2)$ and $C_6^A(q^2)$, and the pion-nucleon-$\Delta$ form factor $G_{\pi N\Delta}(q^2)$. The nucleon form factors are evaluated in the quenched theory and with two degenerate flavors of dynamical Wilson fermions. The nucleon to $\Delta$ form factors, besides Wilson fermions, are evaluated using domain wall valence fermions with staggered sea quark configurations for pion masses as low as about 350 MeV. Using these form factors, together with an evaluation of the renormalized quark mass, we investigate the validity of the diagonal and non-diagonal Goldberger-Treiman relations. The ratios $G_{\pi N\Delta}(q^2)/G_{\pi NN}(q^2)$ and $2C_5^A(q^2)/G_A(q^2)$ are constant as a function of the momentum transfer squared and show almost no dependence on the quark mass. We confirm equality of these two ratios consistent with the Goldberger-Treiman relations extracting a mean value of $1.61(2)$.